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arXiv · 2604.00049

A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations

Jeffrey Uhlmann
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Abstract

A new generalized matrix inverse is derived which is consistent with respect to arbitrary nonsingular diagonal transformations, e.g., it preserves units associated with variables under state space transformations, thus providing a general solution to a longstanding open problem relevant to a wide variety of applications in robotics, tracking, and control systems. The new inverse complements the Drazin inverse (which is consistent with respect to similarity transformations) and the Moore-Penrose inverse (which is consistent with respect to unitary/orthonormal transformations) to complete a trilogy of generalized matrix inverses that exhausts the standard family of analytically-important linear system transformations. Results are generalized to obtain unit-consistent and unit-invariant matrix decompositions and examples of their use are described.

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Jeffrey Uhlmann. 2026-03-30. A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations. https://arxiv.org/abs/2604.00049

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